The bandgap reference voltage circuit is widely used in various applications for providing a stable voltage reference.
As shown in FIG. 1, an example of bandgap reference voltage circuit comprises a first npn bipolar transistor 4, in diode connection, whose emitter terminal is grounded whereas the collector terminal is connected with an end of a first resistor 1. The first resistor 1 has the other end connected with a positive input node of an operational amplifier 6 and with an end of a second resistor 2. The second resistor 3 has the other end connected to the output node 7 of the operational amplifier 6 and an end of a third resistor 3 that has the other end connected to a negative input node of the operational amplifier 6 and the collector of a second npn bipolar transistor 5. The voltage VBG at the output node 7 of the operational amplifier 6 is given by the sum of a base-emitter voltage of the second npn bipolar transistor 5 and the voltage across the third resistor 3, that is:
      V    BG    =            V              BE        ⁢                                  ⁢        2              +                  V        T            ⁢                        R          ⁢                                          ⁢          2                          R          ⁢                                          ⁢          1                    ⁢      ln      ⁢                                    N            ·            R                    ⁢                                          ⁢          2                          R          ⁢                                          ⁢          3                    
where VT is the thermal voltage, R1, R2 and R3 are resistances of resistors 1, 2 and 3, and N is the area ratio of transistors 4 and 5.
The variation of VBE with temperature is −2.2 mV/, while VT is 0.086 mV/. The values of R1, R2, R3 and N are selected to ensure that VBG remain substantially stable over a range of temperature.